Seems from the very beginning; an act establishing several federal crimes in 1790 [1] - only one year after the Constitution came into force - was titled "An Act for the Punishment of Certain Crimes Against the United States".

EDIT: And the root of this phrase seems to be in the distinction between civil and criminal offenses - an crime is an "offense against the State" [2], where the government has standing to bring the case to court rather than the individuals (if any) specifically harmed. In this context, the usage "offense against the United States" pretty clearly indicates crimes where the United States rather than an individual states has jurisdiction.

[1] https://en.wikipedia.org/wiki/Crimes_Act_of_1790

[2] http://criminal.findlaw.com/criminal-law-basics/the-differen...

> Is there some case law that has determined "offenses against the United States" == Federal crime only?

At least going by the spiritual intent of the Constitution, the United States wasn't so much of a country on its own as a coalition of multiple tiny countries (made it easier for trade, war, debts), much as the member states of the European Union are still treated as independent countries today. The European Union hasn't yet had a 'civil' war to change the public opinion and perception of its governance, though.